Chapter 13 Portfolio Theory with Short Sales Constraints

Updated: May 14, 2021

The portfolio theory described in Chapters 11 and 12 assumed that investors can freely short sell all assets. In many situation, however, investors may not be allowed to short sell some or all assets and/or there may be restrictions on when short sales may occur. In particular, certain types of assets (e.g., mutual funds) cannot be sold short. This has implications for asset allocation in individual retirement accounts. Some institutions, such as pension funds, are prevented from directly short selling assets. Government regulations may prevent certain assets from being sold short, or prevent short sales from occurring at certain times. For example, during the 2008 financial crisis the Securities and Exchange Commission temporarily blocked the short sale of financial stocks. Because of these practical limitations on short sales, it is important to be able to incorporate short sales restrictions into mean-variance portfolio theory and this chapter describes how to do it.

This chapter is organized as follows. Section 13.2 describes the impact of short sales constraints, on risky assets, on efficient portfolios in the simplified two risk asset setting presented in Chapter 11. Section 13.3 discusses the computation of mean-variance efficient portfolios when there are short sales constraints in the general case of \(N\) risky assets as described in Chapter 12. Section 13.5 gives a real-world application of the theory to asset allocation among Vanguard mutual funds.

The examples in this chapter use the IntroCompFinR, PerformanceAnalytics, and quadprog packages. Make sure these packages are installed and loaded in R before replicating the chapter examples.

suppressPackageStartupMessages(library(IntroCompFinR))
suppressPackageStartupMessages(library(PerformanceAnalytics))
suppressPackageStartupMessages(library(quadprog))
options(digits=3)